I would like some calrification about the question of max and min for a function in one variable. My doubts are these: our professor told us that when searching for max and min we have to pay attention at tw things; boundary points and points inside the interval. Also I have wrote this on the notebook: the absolute extrema, if they are internal points, they are also local, while the absolute extrema, if they are boundary points, cannot be local too because when we talk about local extrema we have to take into account a complete neighbourhood of that point, hence we need to look from left and right.
Is this true?
I mean for example say I have a function in $[0, 1]$ such that $f(0) = 3$, $f(1/2) = 5$ and $f(1) = 0$. Then shall I say that only $1/2$ is a local extrema and $1/2$ and $1$ as absolute extrema?
Thank you!
Roughly speaking, yes, the idea is what you said. In general you can fix this problem by defining local minimum and maximum with respect to the 'topology' of the domain of your function. Let me explain better:
Given a function $f\colon A\subseteq \mathbb R\to \mathbb R$, we say that $x=x_0$ is a local point of maximum and $f(x_0)$ is a local maximum for $f$, if, there exists an open neighbourhood $(x_0-\delta, x_0+\delta)$ of $x_0$ such that $f(x)\leq f(x_0)$ for each $x\in A\cap [x_0-\delta, x_0+\delta]$.
Therefore, with this definition, you can have also boundary points that can be local maximum points.
By the way, you have to pay attention that these kind of points do not satisfy Fermat theorem, that says if $f$ in derivable at a local maximum (or minimum) point $x_0$ and $x_0$ is an internal point, then $f'(x_0)=0$.
Since this difficulty, then the hypothesis of Rolle Theorem makes much more sense.